Object's details: Morse theory for normal geodesics in sub-Riemannian manifolds with codimension one distributions

Provider:Akademicka Platforma Czasopism

Description

Title:
Morse theory for normal geodesics in sub-Riemannian manifolds with codimension one distributions
Creator:
Giambò, Roberto;
Giannoni, Fabio;
Piccione, Paolo;
Tausk, Daniel V.
Description:
We consider a Riemannian manifold $(\mathcal M,g)$ and a codimension one distribution $\Delta\subset T\mathcal M$ on $\mathcal M$ which is the orthogonal of a unit vector field $Y$ on $\mathcal M$. We do not make any nonintegrability assumption on $\Delta$. The aim of the paper is to develop a Morse Theory for the sub-Riemannian action functional $E$ on the space of horizontal curves, i.e. everywhere tangent to the distribution $\Delta$. We consider the case of horizontal curves joining a smooth submanifold $\mathcal P$ of $\mathcal M$ and a fixed point $q\in\mathcal M$. Under the assumption that $\mathcal P$ is transversal to $\Delta$, it is known (see [P. Piccione and D. V. Tausk, Variational aspects of the geodesic problem is sub-Riemannian geometry , J. Geom. Phys. 39 (2001), 183–206]) that the set of such curves has the structure of an infinite dimensional Hilbert manifold and that the critical points of $E$ are the so called {\it normal extremals} (see [W. Liu and H. J. Sussmann, Shortest paths for sub-Riemannian metrics on rank–$2$ distribution , Mem. Amer. Math. Soc. 564 (1995)]). We compute the second variation of $E$ at its critical points, we define the notions of $\mathcal P$-Jacobi field, of $\mathcal P$-focal point and of exponential map and we prove a Morse Index Theorem. Finally, we prove the Morse relations for the critical points of $E$ under the assumption of completeness for $(\mathcal M,g)$.
Object availability:
free access
Rights:
Prawa autorskie (c) 2016 Topological Methods in Nonlinear Analysis
Date:
2003.06.01
Type:
info:eu-repo/semantics/article;
info:eu-repo/semantics/publishedVersion;
Recenzowany artykuł
Language:
English
Source:
Topological Methods in Nonlinear Analysis; Vol 21, No 2 (June 2003); 273 - 291;
1230-3429
Publisher:
Nicolaus Copernicus University in Toruń
Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2003.016/9603
Subject:
Morse theory;
sub-Riemannian geometry;
nomal sub-Remannian minimizers;
53C17;
53C22;
58E05;
58E10;
58E25
Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2003.016
Data provider:
Akademicka Platforma Czasopism

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